Jim SmithJim SmithJCHSJCHSSection 7-7Section 7-7

THE LAW OF COSINES ALLOWS US TO SOLVE THE TRIANGLETHE LAW OF COSINES ALLOWS US TO SOLVE THE TRIANGLE

WHEN WE CAN’T USE THE LAW OF SINES. SUPPOSE WE KNOWWHEN WE CAN’T USE THE LAW OF SINES. SUPPOSE WE KNOW

THE 3 SIDES OF A TRIANGLE BUT NONE OF THE ANGLES.THE 3 SIDES OF A TRIANGLE BUT NONE OF THE ANGLES.

THE LAW OF COSINES WILL HELP US FIND THE ANGLES.THE LAW OF COSINES WILL HELP US FIND THE ANGLES.

aa

bbcc

AA

CCBB

aa² = b² + c² = b² + c² - 2bc cos A² - 2bc cos Ab² = a² + c² - 2ac cos Bb² = a² + c² - 2ac cos Bc² = a² + b² - 2ab cos Cc² = a² + b² - 2ab cos C

NOTICE WHEN YOU START WITH SIDE NOTICE WHEN YOU START WITH SIDE aa YOU END WITH ANGLE YOU END WITH ANGLE AA

WHEN YOU START WITH SIDE WHEN YOU START WITH SIDE bb YOU END WITH ANGLE YOU END WITH ANGLE BB

WHEN YOU START WITH SIDE WHEN YOU START WITH SIDE cc YOU END WITH ANGLE YOU END WITH ANGLE CC

BE CAREFUL WHEN YOU PUT THIS IN YOUR CALCULATORBE CAREFUL WHEN YOU PUT THIS IN YOUR CALCULATOR

a² = b² + c² - 2bc cos Aa² = b² + c² - 2bc cos A

a = a = (( b² + c² - 2bc cos A b² + c² - 2bc cos A

2 SIDES AND THE INCLUDED ANGLE ( SAS )2 SIDES AND THE INCLUDED ANGLE ( SAS )AA

BBcc

6688

aa

FIND SIDE aFIND SIDE a

a² = b² + c² - 2bc cos Aa² = b² + c² - 2bc cos A

a²a² = 8² + 6² - 2 ( 8 ) ( 6 ) cos 50 = 8² + 6² - 2 ( 8 ) ( 6 ) cos 50

a = ( 8² + 6² - 2 ( 8 ) ( 6 ) cos 50a = ( 8² + 6² - 2 ( 8 ) ( 6 ) cos 50

a = 6.2a = 6.2

5050°°

3 SIDES ( SSS )3 SIDES ( SSS )

AA

BB

CC1010

1515

2222

FIND ANGLE AFIND ANGLE A

a² = b² + c² - 2bc cos Aa² = b² + c² - 2bc cos A

1515² = 22² = 22² + 10² - 2 ( 22 ) ( 10 ) COS A² + 10² - 2 ( 22 ) ( 10 ) COS A

225 = 484 + 100 – 440 COS A225 = 484 + 100 – 440 COS A

225 = 584 – 440 COS A225 = 584 – 440 COS A

-359 = - 440 COS A359 = - 440 COS A

-359359 = COS A = COS A

-440-440

22ndnd COS COS ( ( -359 / -440-359 / -440

A = 35.3A = 35.3°°

Golf On a 180-yard hole, a golfer hits the ball Golf On a 180-yard hole, a golfer hits the ball 175 yards, 3° off-line. How far is the ball from the cup?  175 yards, 3° off-line. How far is the ball from the cup?  A.10.6 yd C.42.1 ydA.10.6 yd C.42.1 ydB.15.0 yd   D.12.2 ydB.15.0 yd   D.12.2 yd

180180

175175

33°°XX

SASSAS

aa² = b² = b² + c² - 2 bc cos 3² + c² - 2 bc cos 3

a² = 180² + 175² - 2( 180 ) ( 175 ) cos 3a² = 180² + 175² - 2( 180 ) ( 175 ) cos 3

a = 180² + 175² - 2( 180 ) ( 175 ) cos 3a = 180² + 175² - 2( 180 ) ( 175 ) cos 3

a = 10.55a = 10.55

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